Topology and self-assembly of defect-colloidal superstructure in confined chiral nematic liquid crystals.

We describe formation of defect-colloidal superstructures induced by microspheres with normal surface anchoring dispersed in chiral nematic liquid crystals in confinement-unwound homeotropic cells. Using three-dimensional nonlinear optical imaging of the director field, we demonstrate that some of the induced defects have nonsingular solitonic nature while others are singular point and line topological defects. The common director structures induced by individual microspheres have dipolar symmetry. These topological dipoles are formed by the particle and a hyperbolic point defect (or small disclination loop) of elementary hedgehog charge opposite to that of a sphere with perpendicular boundary conditions, which in cells with thickness over equilibrium cholesteric pitch ratio approaching unity are additionally interspaced by a looped double-twist cylinder of continuous director deformations. The long-range elastic interactions are probed by holographic optical tweezers and videomicroscopy, providing insights to the physical underpinnings behind self-assembled colloidal structures entangled by twisted solitons. Computer-simulated field and defect configurations induced by the colloidal particles and their assemblies, which are obtained by numerically minimizing the Landau-de Gennes free energy, are in agreement with the experimental findings.

Topological switching and orbiting dynamics of colloidal spheres dressed with chiral nematic solitons.

Metastable configurations formed by defects, inclusions, elastic deformations and topological solitons in liquid crystals are a promising choice for building photonic crystals and metamaterials with a potential for new optical applications. Local optical modification of the director or introduction of colloidal inclusions into a moderately chiral nematic liquid crystal confined to a homeotropic cell creates localized multistable chiral solitons. Here we induce solitons that "dress" the dispersed spherical particles treated for tangential degenerate boundary conditions, and perform controlled switching of their state using focused optical beams. Two optically switchable distinct metastable states, toron and hopfion, bound to colloidal spheres into structures with different topological charges are investigated. Their structures are examined using Q-tensor based numerical simulations and compared to the profiles reconstructed from the experiments. A topological explanation of observed multistability is constructed.

Self-assembly of skyrmion-dressed chiral nematic colloids with tangential anchoring.

We describe dipolar nematic colloids comprising mutually bound solid microspheres, three-dimensional skyrmions, and point defects in a molecular alignment field of chiral nematic liquid crystals. Nonlinear optical imaging and numerical modeling based on minimization of Landau-de Gennes free energy reveal that the particle-induced skyrmions resemble torons and hopfions, while matching surface boundary conditions at the interfaces of liquid crystal and colloidal spheres. Laser tweezers and videomicroscopy reveal that the skyrmion-colloidal hybrids exhibit purely repulsive elastic pair interactions in the case of parallel dipoles and an unexpected reversal of interaction forces from repulsive to attractive as the center-to-center distance decreases for antiparallel dipoles. The ensuing elastic self-assembly gives rise to colloidal chains of antiparallel dipoles with particles entangled by skyrmions.

Chiral bipolar colloids from nonchiral chromonic liquid crystals.

We demonstrate that high anisotropy of elastic constants of chromonic liquid crystals leads to a number of spontaneously twisted nematic director fields around colloidal particles in these non-chiral fluids. For spherical colloidal particles with surface inducing degenerate planar nematic ordering we observe that boojum defects at the particles' poles acquire twisted internal structure, extending up to three particle diameters along the rubbing direction of the cell. The twist handedness of the two boojum defects at the poles of the particle can be either the same or opposite, and we can switch the defects handedness by localized thermal microquenching. Numerical simulations confirm that the transitions into the distorted states are induced by lowering of the twist elastic constant, which results in two (meta)stable chiral configurations of the boojums, separated by an energy barrier much higher than the thermal energy. We show that boojum handedness can change the pairwise elastic interaction between the two particles positioned along the rubbing direction from repulsive to attractive.

Chirality screening and metastable states in chiral nematic colloids.

We show that forces between two colloidal particles in a thin layer of a chiral nematic liquid crystal strongly depend on the chirality of the liquid crystal. The observed pair potentials are attractive, but are oscillatory functions of colloidal separation. The number and the position of local energy minima increase with increasing chirality. The pair interaction is the strongest for the pitch equal to the colloidal diameter and decreases with increasing chirality. We show that the chirality of the medium is responsible for this oscillatory nature and screening of the colloidal interaction in the far and near field. The measurements are in agreement with numerical calculations using Landau-de Gennes theory.

Assembly and control of 3D nematic dipolar colloidal crystals.

Topology has long been considered as an abstract mathematical discipline with little connection to material science. Here we demonstrate that control over spatial and temporal positioning of topological defects allows for the design and assembly of three-dimensional nematic colloidal crystals, giving some unexpected material properties, such as giant electrostriction and collective electro-rotation. Using laser tweezers, we have assembled three-dimensional colloidal crystals made up of 4 µm microspheres in a bulk nematic liquid crystal, implementing a step-by-step protocol, dictated by the orientation of point defects. The three-dimensional colloidal crystals have tetragonal symmetry with antiparallel topological dipoles and exhibit giant electrostriction, shrinking by 25-30% at 0.37 V µm(-1). An external electric field induces a reversible and controllable electro-rotation of the crystal as a whole, with the angle of rotation being ~30° at 0.14 V µm(-1), when using liquid crystal with negative dielectric anisotropy. This demonstrates a new class of electrically highly responsive soft materials.

Colloidal entanglement in highly twisted chiral nematic colloids: twisted loops, Hopf links, and trefoil knots.

The topology and geometry of closed defect loops is studied in chiral nematic colloids with variable chirality. The colloidal particles with perpendicular surface anchoring of liquid crystalline molecules are inserted in a twisted nematic cell with the thickness that is only slightly larger than the diameter of the colloidal particle. The total twist of the chiral nematic structure in cells with parallel boundary conditions is set to 0, π, 2π, and 3π, respectively. We use the laser tweezers to discern the number and the topology of the -1/2 defect loops entangling colloidal particles. For a single colloidal particle, we observe that a single defect loop is winding around the particle, with the winding pattern being more complex in cells with higher total twist. We observe that colloidal dimers and colloidal clusters are always entangled by one or several -1/2 defect loops. For colloidal pairs in π-twisted cells, we identify at least 17 different entangled structures, some of them exhibiting linked defect loops-Hopf link. Colloidal entanglement is even richer with a higher number of colloidal particles, where we observe not only linked, but also colloidal clusters knotted into the trefoil knot. The experiments are in good agreement with numerical modeling using Landau-de Gennes theory coupled with geometrical and topological considerations using the method of tetrahedral rotation.

Early stage domain coarsening of the isotropic-nematic phase transition.

We study numerically the early stage domain coarsening dynamics of the temperature driven isotropic-nematic (I-N) liquid crystal phase transition. System of rod like objects which interact via the modified Lebwohl-Lasher pairwise interaction is considered in 3D. The coarsening dynamics is followed using Brownian molecular dynamics. The box-restricted lattice point fluctuations are allowed in order to get rid of lattice geometry enforced phenomena. We analyze order parameter growth and domain coarsening in the early regime of the I-N phase transition as a function of the quench rate. We show that soon after the transition bimodal distribution of domains appears, where the shorter branch gradually vanishes. The behavior of the system is in accordance with predictions of the Kibble-Zurek mechanism which was originally introduced to model conditions in the early universe.

Light-driven oscillations of entangled nematic colloidal chains.

Laser tweezers have been used to drive the oscillations of a chain of entangled colloidal particles in the nematic liquid crystal 5CB. The amplitude and phase of light-driven oscillations have been determined for the motion of individual colloidal particles. The collective motion of 4.8µm silica particles is highly damped for a driving frequency above 0.5Hz. The results were compared to an effective bead-spring model, where the motion of elastically coupled particles is hindered by viscous damping and hydrodynamic coupling. Qualitative agreement between theory and experiment was obtained.

Calorimetric study of the Paranematic-to-Nematic transition of polydomain side-chain liquid-crystalline elastomers with different mesogen composition.

The phase transition behaviour of various nematic side-chain liquid-crystalline elastomers with different mesogen composition has been explored by means of high-resolution ac calorimetry. Polydomain samples of the same crosslinking density and different type of mesogens have been investigated. The results show a strong dependence of the phase transition features upon the composition of the mesogen. The distance from the critical point, reflected in the sharpness of the heat capacity anomalies, increases when adding a shorter-length mesogen. The results provide new insight for the impact of mesogens on the thermodynamic behaviour and, thus, on the thermomechanical response of nematic liquid-crystalline elastomers.

Vortexlike topological defects in nematic colloids: chiral colloidal dimers and 2D crystals.

We show that chiral ordering of the underlying complex fluid strongly influences defect formation and colloidal interactions. Nonsingular defect loops with a topological charge -2 are observed, with a cross section identical to hyperbolic vortices in magnetic systems. These loops are binding spontaneously formed pairs of colloidal particles and dimers, which are chiral objects. Chiral dimer-dimer interaction weakly depends on the chirality of dimers and leads to the assembly of 2D nematic colloidal crystals of pure or "mixed" chirality, intercalated with a lattice of nonsingular vortexlike defects.

Hierarchical self-assembly of nematic colloidal superstructures.

We show that colloidal superstructures could be assembled in mixtures of large and small colloidal particles dispersed in a nematic liquid crystal. Using elastic interaction of small colloidal particles with the disclination lines we succeed to demonstrate how one can decorate with small particles a topological matrix of defect rings and loops formed by an array of large colloidal particles. Our simulations show that this concept of colloidal self-assembly in nematics could be extended down to the nanoscale particles.

Interactions of quadrupolar nematic colloids.

We present experimental and theoretical study of colloidal interactions in quadrupolar nematic liquid crystal colloids, confined to a thin planar nematic cell. Using the laser tweezers, the particles have been positioned in the vicinity of other colloidal particles and their interactions have been determined using particle tracking video microscopy. Several types of interactions have been analyzed: (i) quadrupolar pair interaction, (ii) the interaction of an isolated quadrupole with a quadrupolar chain, and (iii) the interaction of an isolated quadrupolar colloidal particle with a two-dimensional (2D) quadrupolar crystallite. In all cases, the interactions are of the order of several 100k(B)T for 2 microm particles, which gives rise to relatively stable 2D colloidal crystals. The experimental results are compared to the predictions of Landau-de Gennes theory and we find a relatively good qualitative agreement.

2D interactions and binary crystals of dipolar and quadrupolar nematic colloids.

In this Letter, we demonstrate that the symmetry of the elastic interaction between the dipolar and quadrupolar colloidal particles in the nematic liquid crystal leads to a novel variety of 2D nematic "binary" colloidal crystals, which have not been observed in any colloidal system. The dipolar-quadrupolar interaction is highly anisotropic and shows a power-law dependence when the particles approach each other along the director field with a pair-binding energy of the order of several thousands of k(B)T for 4 microm diameter colloids.

Electrically tunable diffraction of light from 2D nematic colloidal crystals.

We show that diffraction of visible light from 2D dipolar nematic colloidal crystals can be tuned electrically. When the external electric field of approximately 1 V/microm is applied in a direction perpendicular to the plane of the 2D colloidal crystal, the induced strain is highly anisotropic, and the inter-colloidal spacing changes by as much as 20% along one direction and approximately 2% along the perpendicular one. Although the speed of response is in the range of several seconds, this novel mechanism could provide interesting photonic applications.

Entangled nematic colloidal dimers and wires.

It has been predicted, but never confirmed, that colloidal particles in a nematic liquid crystal could be self-assembled by delocalized topological defects and entangled disclinations. We show experimentally and theoretically that colloidal dimers and 1D structures bound by entangled topological defect loops can indeed be created by locally thermally quenching a thin layer of the nematic liquid crystal around selected colloidal particles. The topological entanglement provides a strong stringlike binding, which is ten thousand times stronger compared to water-based colloids. This unique binding mechanism could be used to assemble resonator optical waveguides and robust chiral and achiral structures of topologically entangled colloids that we call colloidal wires.

Two-dimensional dipolar nematic colloidal crystals.

We study the interactions and directed assembly of dipolar nematic colloidal particles in planar nematic cells using laser tweezers. The binding energies for two stable configurations of a colloidal pair with homeotropic surface alignment are determined. It is shown that the orientation of the dipolar colloidal particle can efficiently be controlled and changed by locally quenching the nematic liquid crystal from the laser-induced isotropic phase. The interaction of a single colloidal particle with a single colloidal chain is determined and the interactions between pairs of colloidal chains are studied. We demonstrate that dipolar colloidal chains self-assemble into the two-dimensional (2D) dipolar nematic colloidal crystals. An odd-even effect is observed with increasing number of colloidal chains forming the 2D colloidal crystal.

Liquid crystal 8CB in random porous glass: NMR relaxometry study of molecular diffusion and director fluctuations.

We present the measurements of the proton spin-lattice relaxation time T1 of liquid crystal 4-n-octyl-4'-cyanobiphenyl (8CB) confined into randomly oriented approximately 15 nm pores of untreated porous glass. In the low kilohertz range the spin-lattice relaxation rate in the nanoconfined 8CB is about ten times larger than in the bulk. We show that the increase is mainly due to molecular reorientations mediated by translational displacements (RMTD). In the paranematic phase the power law describing the RMTD dispersion, (T1(-1))RMTD proportional, omega(-p), is well characterized by the exponent p=0.5+/-0.06 and suggests an equipartition of diffusion modes with different wavelengths. The largest distance related to the decay of the orientational correlation function is about twice the diameter of the cavity. The situation is different in the nematic phase, where the orientational correlation is eventually lost at approximately 60 nm in the direction along the pore, a distance corresponding roughly to the length of a pore segment in the glassy matrix. The exponent p is between 0.65 and 0.9, depending on the temperature, which implies that in the nematic phase long wavelength modes are relatively more important--a consequence of the uniform director field along the pore. These observations are in agreement with the model of mutually independent pores with nematic director parallel to the pore axis in each segment. We point out that in strongly confined liquid crystals the proton NMR relaxometry does not provide the evidence of director fluctuations correlated over micrometer distances as was suggested earlier. The local translational diffusion of molecules within the cavities is found about as fast as in bulk.

Annihilation of nematic point defects: pre-collision and post-collision evolution.

The annihilation of the nematic hedgehog and anti-hedgehog within an infinite cylinder of radius R is studied. The semi-microscopic lattice-type model and Brownian molecular dynamics are used. We distinguish among the i) early pre-collision, ii) late pre-collision, iii) early post-collision, and iv) late post-collision stages. In the pre-collision stage our results agree qualitatively with the existing experimental observations and also continuum-type simulations. The core of each defect exhibits a ring-like structure, where the ring axis is set perpendicular to the cylinder symmetry axis. For xi(0)d/(2R) > 1 the interaction between defects is negligible, where xi(0)d describes the initial separation of defects. Consequently, the defects annihilate within the simulation time window for xi(0)d/(2R) < 1. For close enough defects their separation scales as xi(d) [see text] (t(c)- t)(0.4+/-0.1), where t(c) stands for the collision time. In elastically anisotropic medium the hedgehog is faster than the anti-hedgehog. In the early pre-collision stage the defects can be treated as point-like particles, possessing inherent core structure, that interact via the nematic director field. In the late pre-collision stage the cores reflect the interaction between defects. After the collision a charge-less ring structure is first formed. In the early post-collision stage the ring adopts an essentially untwisted circular structure of the radius xi(r). In the late post-collision stage we observe two qualitatively different scenarios. For mu = xi(r)/R < mu(c) approximately 0.25 the ring collapses leading to the escaped radial equilibrium structure. For mu > mu(c) the chargeless ring triggers the nucleation growth into the planar polar structure with line defects.

Laser trapping of low refractive index colloids in a nematic liquid crystal.

We describe and analyze laser trapping of small colloidal particles in a nematic liquid crystal, where the index of refraction of colloids is smaller compared to the indices of the liquid crystal. Two mechanisms are identified that are responsible for this anomalous trapping: (i) below the optical Fréedericksz transition, the trapping is due to the anisotropic dielectric interaction of the polarized light with the inhomogeneous director field around the colloid, (ii) above the optical Fréedericksz transition, the optical trapping is accompanied by the elasticity-mediated interaction between the optically distorted region of a liquid crystal and the colloid. In the majority of the experiments, the trapping above the Fréedericksz transition is highly anisotropic. Qualitative agreement is found with a numerical analysis, considering the nematic director elastic distortion, dielectric director-light field coupling and optical repulsion due to low refraction index colloid in high index surroundings.